Why does lim_{n\to\infty} work in my LaTeX document?

The macro for the limit operator is \lim.

Without the \, it just treated as three characters l, i, m. This is no different that $xy$ representing a product of two terms x, and y, so $lim$ is a product of three terms: l, i, m. So with $lim_{n\to\infty}$, the subscript is applied to the m term. Perhaps the meaning is more obvious if you write and equivalent statement:

$ l  i  m_{n\to\infty}$

Note that without the \ the three letters are in italics, representing variables. The operator \lim is not in italics representing an operator.

\documentclass[fleqn]{article}
\begin{document}
$lim_{n\to\infty}$

$\lim_{n\to\infty}$

\[ lim_{n\to\infty} \]
\[\lim_{n\to\infty} \]
\end{document}